Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5915,2,Mod(1,5915)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5915, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5915.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 5915 = 5 \cdot 7 \cdot 13^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 5915.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | yes |
Analytic conductor: | \(47.2315127956\) |
Analytic rank: | \(0\) |
Dimension: | \(21\) |
Twist minimal: | yes |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 | −2.54595 | 0.867067 | 4.48187 | 1.00000 | −2.20751 | 1.00000 | −6.31871 | −2.24819 | −2.54595 | ||||||||||||||||||
1.2 | −2.53402 | 0.274221 | 4.42125 | 1.00000 | −0.694882 | 1.00000 | −6.13550 | −2.92480 | −2.53402 | ||||||||||||||||||
1.3 | −2.06819 | 0.524302 | 2.27742 | 1.00000 | −1.08436 | 1.00000 | −0.573763 | −2.72511 | −2.06819 | ||||||||||||||||||
1.4 | −1.76432 | −2.79517 | 1.11284 | 1.00000 | 4.93158 | 1.00000 | 1.56524 | 4.81295 | −1.76432 | ||||||||||||||||||
1.5 | −1.42081 | 3.33012 | 0.0187087 | 1.00000 | −4.73148 | 1.00000 | 2.81504 | 8.08969 | −1.42081 | ||||||||||||||||||
1.6 | −1.32241 | 2.15086 | −0.251224 | 1.00000 | −2.84433 | 1.00000 | 2.97705 | 1.62621 | −1.32241 | ||||||||||||||||||
1.7 | −1.05806 | −1.50254 | −0.880506 | 1.00000 | 1.58978 | 1.00000 | 3.04775 | −0.742381 | −1.05806 | ||||||||||||||||||
1.8 | −0.872229 | −1.78534 | −1.23922 | 1.00000 | 1.55722 | 1.00000 | 2.82534 | 0.187428 | −0.872229 | ||||||||||||||||||
1.9 | −0.381801 | 2.01941 | −1.85423 | 1.00000 | −0.771013 | 1.00000 | 1.47155 | 1.07803 | −0.381801 | ||||||||||||||||||
1.10 | −0.136571 | 0.775570 | −1.98135 | 1.00000 | −0.105921 | 1.00000 | 0.543737 | −2.39849 | −0.136571 | ||||||||||||||||||
1.11 | 0.449582 | −0.613470 | −1.79788 | 1.00000 | −0.275805 | 1.00000 | −1.70746 | −2.62366 | 0.449582 | ||||||||||||||||||
1.12 | 0.826316 | −3.03579 | −1.31720 | 1.00000 | −2.50852 | 1.00000 | −2.74106 | 6.21602 | 0.826316 | ||||||||||||||||||
1.13 | 0.952302 | −2.74726 | −1.09312 | 1.00000 | −2.61623 | 1.00000 | −2.94559 | 4.54746 | 0.952302 | ||||||||||||||||||
1.14 | 1.06682 | 3.07881 | −0.861896 | 1.00000 | 3.28453 | 1.00000 | −3.05313 | 6.47906 | 1.06682 | ||||||||||||||||||
1.15 | 1.37365 | 1.75471 | −0.113079 | 1.00000 | 2.41036 | 1.00000 | −2.90264 | 0.0789971 | 1.37365 | ||||||||||||||||||
1.16 | 1.55527 | −1.25293 | 0.418865 | 1.00000 | −1.94865 | 1.00000 | −2.45909 | −1.43016 | 1.55527 | ||||||||||||||||||
1.17 | 2.01969 | −2.51514 | 2.07915 | 1.00000 | −5.07981 | 1.00000 | 0.159865 | 3.32594 | 2.01969 | ||||||||||||||||||
1.18 | 2.03843 | 2.35682 | 2.15519 | 1.00000 | 4.80422 | 1.00000 | 0.316346 | 2.55462 | 2.03843 | ||||||||||||||||||
1.19 | 2.52854 | 3.10549 | 4.39352 | 1.00000 | 7.85236 | 1.00000 | 6.05211 | 6.64408 | 2.52854 | ||||||||||||||||||
1.20 | 2.54977 | 1.71797 | 4.50132 | 1.00000 | 4.38043 | 1.00000 | 6.37779 | −0.0485752 | 2.54977 | ||||||||||||||||||
See all 21 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(-1\) |
\(7\) | \(-1\) |
\(13\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 5915.2.a.bp | yes | 21 |
13.b | even | 2 | 1 | 5915.2.a.bo | ✓ | 21 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
5915.2.a.bo | ✓ | 21 | 13.b | even | 2 | 1 | |
5915.2.a.bp | yes | 21 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5915))\):
\( T_{2}^{21} - 4 T_{2}^{20} - 23 T_{2}^{19} + 105 T_{2}^{18} + 200 T_{2}^{17} - 1137 T_{2}^{16} - 755 T_{2}^{15} + 6612 T_{2}^{14} + 525 T_{2}^{13} - 22578 T_{2}^{12} + 5415 T_{2}^{11} + 46557 T_{2}^{10} - 19744 T_{2}^{9} - 57394 T_{2}^{8} + \cdots - 125 \) |
\( T_{3}^{21} - 5 T_{3}^{20} - 33 T_{3}^{19} + 193 T_{3}^{18} + 397 T_{3}^{17} - 3073 T_{3}^{16} - 1722 T_{3}^{15} + 26077 T_{3}^{14} - 4484 T_{3}^{13} - 127331 T_{3}^{12} + 76017 T_{3}^{11} + 361154 T_{3}^{10} - 314954 T_{3}^{9} + \cdots + 8128 \) |